Question

Write an exponential function that models the points in the table remember your function should be write y-a(b)^x
X:0,1,2,3
Y:80,20,5,1.25

Answer 1

To find an exponential function of the form \( y = a(b)^x \) that fits the given points, we can first look for the values of \( a \) and \( b \).

Given the table of values:
- \( (0, 80) \)
- \( (1, 20) \)
- \( (2, 5) \)
- \( (3, 1.25) \)

1. **Identify the value of \( a \)**:
The value of \( a \) is the \( y \)-value when \( x = 0 \):
\[
a = 80
\]

2. **Determine the value of \( b \)**:
We can use the ratio of consecutive \( y \)-values to find \( b \):

From \( x = 0 \) to \( x = 1 \):
\[
\frac{y(1)}{y(0)} = \frac{20}{80} = \frac{1}{4}
\]

From \( x = 1 \) to \( x = 2 \):
\[
\frac{y(2)}{y(1)} = \frac{5}{20} = \frac{1}{4}
\]

From \( x = 2 \) to \( x = 3 \):
\[
\frac{y(3)}{y(2)} = \frac{1.25}{5} = \frac{1.25}{5} = \frac{1}{4}
\]

Since the ratio of consecutive \( y \)-values is consistently \( \frac{1}{4} \), we can determine that:
\[
b = \frac{1}{4}
\]

3. **Write the function**:
Using the values of \( a \) and \( b \) we found, we can now write our exponential function:
\[
y = 80\left( \frac{1}{4} \right)^x
\]

Thus, the exponential function that models the points in the table is:
\[
y = 80 \left( \frac{1}{4} \right)^x
\]